Absolute Infinitive

Hi.

About the Absolute Infinitive.

In Smyth 1247. on Absolute Infinitive, there are sample sentences like,

hekousa einai ouk apoleipetai (it is not willingly separated
to ge ep’ ekeinon einai eso^the^s (as far as it depended on him you would have been saved)

Can I understand them as being adverbial, like ho^s clauses, explaining the accompanying circumstance of the main clause ?

Then,

antelabeto Israe^l paidos autou,
mne^sthe^nai eleous
(he helped Israel his slaves,…)

Is this “mne^sthe^nai” such an adverbial expression ?
“(with the accompanying circumstance of) remembering the pitious people” ?